Abstract (inglese)

Let V be a complete discrete valuation ring of mixed characteristic (0,p), K be the fraction field and k be the residue field. We study p-adic differential equations on a semistable variety over V.
We consider a proper semistable variety X over V and a relative normal crossing divisor D on it.
We consider on X the open U defined by the complement of the divisor D and we call U_K and U_k the generic fiber and the special fiber respectively. In an analogous way we call D_K, X_K and D_k, X_k the generic and the special fiber of D, X.
In the geometric situation described, we investigate the relations between algebraic differential equations on X_K and analytic differential equations on the rigid analytic space associated to the completion of X along its special fiber.

The main result is the existence and the full faithfulness of an algebrization functor between the following categories:
1) the category of locally free overconvergent log isocrystals on the log pair (U_k,X_k), (where the log structure is defined by the divisor given by the union of X_k e D_k), with unipotent monodromy;
2) the category of modules with connection on U_K, regular along D_K, which admit an extension to modules with connection on X_K with nilpotent residue.